Construction of Fractional Power Series Solutions to Fractional Boussinesq Equations Using Residual Power Series Method

This paper is aimed at constructing fractional power series (FPS) solutions of time-space fractional Boussinesq equations using residual power series method (RPSM). Firstly we generalize the idea of RPSM to solve any-order time-space fractional differential equations in high-dimensional space with initial value problems inR. Using RPSM, we can obtain FPS solutions of fourth-, sixth-, and 2nth-order time-space fractional Boussinesq equations in R and fourth-order time-space fractional Boussinesq equations in R2 and R. Finally, by numerical experiments, it is shown that RPSM is a simple, effective, and powerful method for seeking approximate analytic solutions of fractional differential equations.


Introduction
The study of nonlinear problems is of crucial importance not only in mathematics but also in physics, engineering, economic, and other disciplines, since most phenomena in our world are essentially nonlinear.But it is very difficult to establish mathematical models for nonlinear problem.In the field of engineering and science, we come across physical and natural phenomena which are represented by mathematical models, happening to differential equations.For example, simple harmonic motion, equation of motion, and deflection of a beam are represented by differential equations.Hence, the behaviour of differential equations is a necessity in such studies.A natural idea is to seek the exact solution, but it is often too difficult to succeed.So, as in [1], we attempt to construct approximate solution.Many assumptions have to be made artificially or unnecessarily in the integer-order differential system to make the practical problems solvable, leading to loss of most significant information.So it is necessary to construct new systems to solve these problems and fractional-order system is one of these.That is to say, fractional differential system is much well suited to physical problems compared to its differential partner since it makes less unnecessary or overrestricted assumption which may change the problem being solved, sometimes seriously [2].
The subject of fractional-order system dates back from the idea of derivatives of noninteger order which initially appeared in a letter from Leibnitz to L'Hospital in 1695 about the notation   /  .L'Hospital posed a question to Leibnitz: "what would the result be if  = 1/2?"Leibnitz replied [3] "it follows that  1/2  will be equal to  √  : .This is an apparent paradox, from which, one day useful consequences will be drawn."From these words, fractional calculus was born.And so most authors on this topic will cite a particular date (September 30, 1695) as the birthday of the so-called "fractional calculus" [4].However, at that time, there are little specific models based on this kind of derivative, so the study of fractional order system attracts little attention.
Nowadays, more and more researchers pay massive attention to fractional calculus and they find that fractional systems can seize the missed information in the integral order systems.It is well known that the integer-order differential operators and the integer-order integral operators are local, but on the aspect as well, the fractional-order differential operators and the fractional-order integral operators are nonlocal operators.This means that the next state of a system 2 Mathematical Problems in Engineering depends not only upon its current state but also upon all of its historical states.In fact, this is the main reason why differential operators of fractional order provide an excellent instrument for description of memory and hereditary properties of various mathematical, physical, and engineering processes [5,6].More details about the fractional differential theory can be found in [7][8][9][10][11][12][13].
Particularly, power series has played an important role in the study of elementary functions in the theory of calculus and it has been widely used in computational science for easily obtaining an approximation of functions.In physics, chemistry, and many other sciences, this power series expansion has allowed scientists to make an approximate study of many systems, neglecting higher-order terms around the equilibrium point.This is a fundamental tool to linearize a problem, which guarantees easy analysis [1,6,[14][15][16].
The motivation of this paper is to study the time-space fractional Boussinesq equations.The motion of water waves plays an important role in oceanographic engineering and for most geographical areas, which are produced by environmental actions on beaches or on man-made fixed or floating structures.Natural wave trains are irregular in shape and they interact because the propagation process is nonlinear.Classical wave theory failed to describe the combined effect of these processes and they can be divided into two categories: (1) a nonlinear description of monochromatic waves of a specific frequency or wave length; (2) a linear description of irregular waves based on superposition of individual frequency components with random phases [17,18].
So many mathematicians and physicists tried to contract new model to describe the process of wave propagation on the surface of water.In 1872, the French scientist Joseph Boussinesq (1842-1929) derived a model equation [18] for the propagation of water waves from Euler's equations to describe the propagation process of long waves on the surface of water with small amplitude.This equation was formulated for analysis of long waves in shallow water.Later the general form of Boussinesq equation [17] was applied to problems not only in the study of the dynamics of thin inviscid layers with free surface but also in the study of nonlinear string, the shape-memory alloys, the propagation of waves in elastic rods, shallow wave waters (waves with long wavelength compared to the depth), and the continuum limit of lattice dynamics or coupled electrical circuits.More and more researchers pay attention to this equation and get a number of interesting results which were applied to many areas [17][18][19][20].In recent years, with the development of fractional system rapidly, the classical Boussinesq equation evolves into the fractional Boussinesq equation, which is suitable for studying the water propagation through heterogeneous porous media [21][22][23][24].This paper is concerned with the following initial value problem for time-space fractional Boussinesq equation in high-dimensional space: where  [23], they obtained the solutions in terms of fractional hypergeometric functions, fractional triangle functions, and a rational function.Different from the solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations in R obtained by Arqub et al. [15], in this paper we attempt to construct the more general approximate fractional power series solutions in R  using RPSM.Furthermore, this method is valid for high-dimensional space and any-order time-space partial differential equations.The organization of the rest of this paper is as follows: in Section 2, an introduction of some concepts on fractional calculus theory is presented.In Section 3, we introduce algorithm of RPSM for any-order time-space fractional differential equations with initial value problems.In Section 4, error analysis of RPSM is discussed simply.In Section 5, we apply RPSM to obtain fractional power series solutions of time-space fractional Boussinesq equations with initial value problems.The paper ends with some simple conclusions.

Concepts on Fractional Calculus Theory
There are several definitions of the fractional integration of order  ≥ 0, not necessarily equivalent to each other; see [9].Riemann-Liouville and Caputo fractional definitions are the two most used from all the other definitions of fractional calculus which have been introduced recently [5,6,[26][27][28][29].
Definition 1.The Mittag-Leffler function is defined as follows: ,  > 0.   Definition 5.For  to be the smallest integer that exceeds , the Caputo time-fractional derivative operator of order  of (, ) is defined as follows: And the space fractional derivative of order  is defined as follows: Definition 6.A power series representation of the form where  − 1 <  <  and  ⩾  0 , is called a fractional power series about  0 , where  is a variable and   are the coefficients of the series.

Algorithm of RPSM
In [14], Abu Arqub et al. give RPSM to obtain power series solution of higher-order ordinary differential equation with initial conditions in R in the integer system.In [5], El-Ajou et al. give the iterative progress of RPSM for time-fractional KdV-Burgers equation in R. In this section, we give the more general RPSM to find out fractional power series solution for any-order time-space fractional differential equations with initial conditions in R  .
Since (, ) is analytic about , the solution of the system can be written in the form of where   (, ) are terms of approximations and given as where  is the radius of convergence of above series.Obviously, when  = 0, 1, 2, . . .,  − 1, since the terms    (, ) satisfy the initial condition, we can get So we have the initial guess approximation of (, ) in the following form: On the other aspect as well, if we choose  initial (, ) as initial guess approximation   (, ) for  = ,  + 1,  + 2, . .., the approximate of the solution of (, ) of ( 13) by the th truncated series is Before applying RPS method for solving (13), we give some notes: Substituting the th truncated series   (, ) into ( 13), we can obtain the following definition for th residual function: where Then we have the following facts: (1) lim →∞   (, ) = (, ); (2) Res(, ) = 0; In fact, the relation of ( 24) is a fundamental rule in RPSM and its applications.So the FPS solution of ( 13) is where   () ( = 0, 1, 2, . . .,  − 1) are given and   () ( = ,  + 1,  + 2, . ..) have been constructed by RPSM in the form (24).The algorithm of RPSM is considered a direct and simple approach because it depends on the recursive differentiation of time-fractional derivative and the use of given initial conditions to calculate coefficients of the multiple FPS solution using minimal calculations.The RPS does not require linearization, perturbation, or discretization of the variables; is not affected by computational round-off errors; and does not require large computer memory and extensive time.
To show the accuracy and efficiency of RPSM, we give two types of error: where Rel  (, ) denotes the relative error.
RPSM provides an approximate analytical solution in terms of an infinite FPS and it is necessary to give the errors of the approximate solution.In the next section, we apply RPSM to time-space fractional Boussinesq equations and take fourth-order time-space fractional Boussinesq equations as an example to illustrate the absolute error and relative error of the approximate solution.

Application of RPSM to Time-Fractional Boussinesq Equations
In this section, the procedure of constructing approximate analytical solutions to time-space fractional Boussinesq equations using RPSM is presented.

Fourth-Order Time-Space
Fractional Boussinesq Equations in R. Consider the fourth-order time-space fractional Boussinesq equation with one-dimensional space variable: where , , , and  are constant coefficients, and  0 () and  1 () ∈  ∞ (R).
Since we choose the special constant coefficients and give the initial value  0 () =   and  1 () = √ 2  , so the exact solution in (36) is easily obtained.However, the initial value in application may be "complex" which leads to difficulty to obtain the exact solution and even we cannot obtain it, so it is necessary to construct the approximate solution by RPSM.Thus it is necessary to illustrate the the validity and reliability of RPSM in order to construct numerical solution for generalized initial value.
In the Appendix, from the comparison of Tables 1-4, if the variable  is given, the absolute error Exa  (, ) and the relative error Rel  (, ) tend to 0 as  tends to 0. So the behaviour of the exact solution can be described by the approximate fractional power series solution around the point (0, 0).

Fourth-Order Time-Space
Fractional Boussinesq Equations in R 2 .Consider the fourth-order time-space fractional Boussinesq equation with two-dimensional space variables: where are constants, and  0 () and  1 () ∈  ∞ (R 2 ).
Before solving this problem, we give several notations for convenience: So system (65) is equal to According to (25), the solution of (67) can be written in the following form: Apparently, according to (17), we have Using ( 18)-( 22), we can obtain which is equal to (74).